Real and Complex Analysis (Rudin)

264 REAL AND COMPLEX ANALYSIS
Exercises
1 Suppose .1. is a closed equilateral triangle in the plane, with vertices a, b, c. Find
max (I z - a I I z - b I I z - c I) as z ranges over .1..
2 Supposefe H(II+), where 11+ is the upper half plane, and If I :s; 1. How large can 1f'(i)I be? Find
the extremal functions. (Compare the discussion in Sec. 12.5.)
3 Supposefe H(Q). Under what conditions can If I have a local minimum in Q?
4 (a) Suppose Q is a region, D is a disc, jj c Q,fe H(Q),fis not constant, and If I is constant on the
boundary of D. Prove thatfhas at least one zero in D.
(b) Find all entire functionsfsuch that I f(z) I = 1 whenever Iz 1= 1.
5 Suppose Q is a bounded region, {In} is a sequence of continuous functions on n which are holomorphic
in Q, and {In} converges uniformly on the boundary of Q. Prove that {In} converges uniformly
on n.
6 Supposefe H(Q), r is a cycle in Q such that Ind r (IX) = 0 for all IX ¢ Q, Ifml :s; 1 for every' e P,
and Ind r (z) # O. Prove that I f(z) I :s; 1.
7 In the proof of Theorem 12.8 it was tacitly assumed that M(a) > 0 and M(b) > o. Show that the
theorem is true if M(a) = 0, and that thenf(z) = 0 for all z e Q.
8 If 0 < Rl < R2 < 00, let A(Rl> R2) denote the annulus
{z: R. < Izl < R2 }.
There is a vertical strip which the exponential function maps onto A(Rl> R 2). Use this to prove
Hadamard's three-circle theorem: Iff e H(A(R 1, R 2)), if
M(r) = max I f(re~ I
and if Rl < a < r < b < R 2 , then
log (b/r)
log (r/a)
log M(r) :s; log (b/a) log M(a) + log (b/a) log M(b).
[In other words, log M(r) is a convex function of log r.] For which f does equality hold in this
inequality?
9 Let II be the open right half plane (z e II if and only if Re z > 0). Suppose f is continuous on the
closure of II (Re z ~ 0),1 e H(II), and there are constants A < 00 and IX < 1 such that
I f(z) I < A exp ( I z I")
for all z e II. Furthermore, I f(iy) I :s; 1 for all real y. Prove that I f(z) I :s; 1 in II.
Show that the conclusion is false for IX = 1.
How does the result have to be modified if II is replaced by a region bounded by two rays
through the origin, at an angle not equal to 7t?
10 Let II be the open right half plane. Suppose that f e H(II), that I f(z) I < 1 for all z e II, and that
there exists IX, -7t/2 < IX < 7t/2, such that
log I f(re 1 ") I
-----"-'-'--'--'-'-+ - 00 as r-+ 00.
r
Prove that f = O.
Hint: Put gn(z) = f(z)e"z, n = 1,2,3, .... Apply Exercise 9 to the two angular regions defined by
-7t/2 < 6 < IX, IX < 6 < 7t/2. Conclude that each gn is bounded in II, and hence that I gn I < 1 in II, for
all n.
11 Suppose r is the boundary of an unbounded region Q,f e H(Q), f is continuous on Q u r, and
there are constants B < 00 and M < 00 such that I f I :s; M on r and I f I :s; B in Q. Prove that we
then actually have I f I :s; M in Q.